ただいま作成中
私の授業で使いながら問題を増やしているため、完成するまでに時間がかかりそうです。少しずつ問題を増やしたり、ポイント解説を付けたりしていきます。無限の彼方で完成する日を、どうぞご期待ください。
Happy Math-ing!
未完成でもよければ、使ってやってください。😃
次の３次方程式を解け。
↓この問題へのリンクはこちら（右クリックで保存）
【解答】
\def\siki{x^3-4x^2}\def\teisuf{+}\def\teisu{8}
\def\yakusu{\pm1,\ \pm2,\ \pm4,\ \pm8}
\def\testsiki{%
& \colMM{orange}{P(1) = 1-4+8=5}\\
& \colMM{orange}{P(-1) = -1-4+8=3}\\
& \colMM{orange}{P(2) = 8-16+8=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた！}
}
\def\dainyu{2}
\def\dainyusiki{%
\begin{align*}
P(\dainyu) &= 2^3-4 \cdot 2^2 +8\\
\colMM{green}{\Darr } &= 8-16+8\\
\colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが０・・・割り切れる！}
\end{align*}
}
\def\warusiki{x-2}
\def\syou{x^2-2x-4}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{align*}
& \colFB{orange}{下調べ}\\
& \ \colMM{orange}{P(k)=0\ になる\ k\ を探す！}\\
& \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入！}\\
\testsiki\\
\\
& P(x) = \siki\teisuf\teisu\ とすると\\
& \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\
& \dainyusiki\\
& \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\
& よって，P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\
& \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り０}\\
& P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\
\\
& P(x) = 0\ から\\
& \warusiki=0 または \syou=0\\
\\
& \begin{align*}
\warusiki &= 0\\
x &= \dainyu
\end{align*}\\
\\
& \syou = 0\\
& \begin{align*}
x &= \dfrac{2 \pm \sqrt{4-4 \cdot (-4)}}{2}\\
&= \dfrac{2 \pm \sqrt{20}}{2}\\
&= \dfrac{2 \pm 2\sqrt{5}}{2}\\
&= \dfrac{2(1 \pm \sqrt{5})}{2} = 1 \pm \sqrt{5}\\
\end{align*}\\
\\
& したがって x = 2,\ 1\pm\sqrt{5}
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan組立除法
\def\va{1}
\def\vbf{}\def\vb{-4}
\def\vcf{+}\def\vc{0}
\def\vdf{+}\def\vd{8}
\def\wv{2}
\def\ws{x-2}
\def\kb{2}
\def\tbf{}\def\tb{-2}
\def\kc{-4}
\def\tcf{}\def\tc{-4}
\def\kd{-8}
\def\td{0}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{matrix}
\va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\
\colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\
& & & & \colMM{red}{\Darr}\\
\va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\
& \kb & \kc & \kd & \\\hline
\va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\
\colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\
\va x^2 & \tb x & \tc\\
\end{matrix}
\\
\begin{align*}\\
\colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\
& \ \colFR{black}{\scriptsize\bf 余} \ \td
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan↓この問題へのリンクはこちら（右クリックで保存）
【解答】
\def\siki{x^3+4x^2+x}\def\teisuf{}\def\teisu{-6}
\def\yakusu{\pm1,\ \pm2,\ \pm3,\ \pm6}
\def\testsiki{%
& \colMM{orange}{P(1) = 1+4+1-6=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた！}
}
\def\dainyu{1}
\def\dainyusiki{%
\begin{align*}
P(\dainyu) &= 1^3+4 \cdot 1^2 +1-6\\
\colMM{green}{\Darr } &= 1+4+1-6\\
\colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが０・・・割り切れる！}
\end{align*}
}
\def\warusiki{x-1}
\def\syou{x^2+5x+6}
\def\insubunkai{(x+2)(x+3)}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{align*}
& \colFB{orange}{下調べ}\\
& \ \colMM{orange}{P(k)=0\ になる\ k\ を探す！}\\
& \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入！}\\
\testsiki\\
\\
& P(x) = \siki\teisuf\teisu\ とすると\\
& \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\
& \dainyusiki\\
& \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\
& よって，P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\
& \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り０}\\
& \begin{align*}
P(x) &= (\warusiki)(\syou)\colMM{magenta}{+0}\\
&= (\warusiki)\insubunkai\\
\end{align*}\\
\\
& P(x) = 0\ から\\
& \warusiki=0{\scriptsize または }x+2=0{\scriptsize または }x+3=0\\
\\
& したがって x = 1,\ -2,\ -3
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan組立除法
\def\va{1}
\def\vbf{+}\def\vb{4}
\def\vcf{+}\def\vc{1}
\def\vdf{}\def\vd{-6}
\def\wv{1}
\def\ws{x-1}
\def\kb{1}
\def\tbf{+}\def\tb{5}
\def\kc{5}
\def\tcf{+}\def\tc{6}
\def\kd{6}
\def\td{0}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{matrix}
\va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\
\colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\
& & & & \colMM{red}{\Darr}\\
\va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\
& \kb & \kc & \kd & \\\hline
\va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\
\colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\
\va x^2 & \tb x & \tc\\
\end{matrix}
\\
\begin{align*}\\
\colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\
& \ \colFR{black}{\scriptsize\bf 余} \ \td
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan↓この問題へのリンクはこちら（右クリックで保存）
【解答】
\def\siki{x^3+4x^2+5x}\def\teisuf{+}\def\teisu{2}
\def\yakusu{\pm1,\ \pm2}
\def\testsiki{%
& \colMM{orange}{P(1) = 1+4+5+2=12}\\
& \colMM{orange}{P(-1) = -1+4-5+2=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた！}
}
\def\dainyu{-1}
\def\dainyusiki{%
\begin{align*}
P(\dainyu) &= (-1)^3+4 \cdot (-1)^2 +5 \cdot (-1)+2\\
\colMM{green}{\Darr } &= -1+4-5+2\\
\colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが０・・・割り切れる！}
\end{align*}
}
\def\warusiki{x+1}
\def\syou{x^2+3x+2}
\def\insubunkai{(x+1)(x+2)\\&=(x+1)^2(x+2)}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{align*}
& \colFB{orange}{下調べ}\\
& \ \colMM{orange}{P(k)=0\ になる\ k\ を探す！}\\
& \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入！}\\
\testsiki\\
\\
& P(x) = \siki\teisuf\teisu\ とすると\\
& \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\
& \dainyusiki\\
& \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\
& よって，P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\
& \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り０}\\
& \begin{align*}
P(x) &= (\warusiki)(\syou)\colMM{magenta}{+0}\\
&= (\warusiki)\insubunkai\\
\end{align*}\\
\\
& P(x) = 0\ から\\
& \warusiki=0{\scriptsize または }x+2=0\\
\\
& したがって x = -1,\ -2\colMM{lightgray}{ （x=-1\ は重解）}
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan組立除法
\def\va{1}
\def\vbf{+}\def\vb{4}
\def\vcf{+}\def\vc{5}
\def\vdf{+}\def\vd{2}
\def\wv{-1}
\def\ws{x+1}
\def\kb{-1}
\def\tbf{+}\def\tb{3}
\def\kc{-3}
\def\tcf{+}\def\tc{2}
\def\kd{-2}
\def\td{0}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{matrix}
\va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\
\colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\
& & & & \colMM{red}{\Darr}\\
\va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\
& \kb & \kc & \kd & \\\hline
\va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\
\colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\
\va x^2 & \tb x & \tc\\
\end{matrix}
\\
\begin{align*}\\
\colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\
& \ \colFR{black}{\scriptsize\bf 余} \ \td
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan↓この問題へのリンクはこちら（右クリックで保存）
【解答】
\def\siki{x^3-3x^2}\def\teisuf{+}\def\teisu{2}
\def\yakusu{\pm1,\ \pm2}
\def\testsiki{%
& \colMM{orange}{P(1) = 1-3+2=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた！}
}
\def\dainyu{1}
\def\dainyusiki{%
\begin{align*}
P(\dainyu) &= 1^3-3 \cdot 1^2+2\\
\colMM{green}{\Darr } &= 1-3+2\\
\colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが０・・・割り切れる！}
\end{align*}
}
\def\warusiki{x-1}
\def\syou{x^2-2x-2}
\def\insubunkai{(x+1)^2}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{align*}
& \colFB{orange}{下調べ}\\
& \ \colMM{orange}{P(k)=0\ になる\ k\ を探す！}\\
& \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入！}\\
\testsiki\\
\\
& P(x) = \siki\teisuf\teisu\ とすると\\
& \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\
& \dainyusiki\\
& \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\
& よって，P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\
& \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り０}\\
& P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\
\\
& P(x) = 0\ から\\
& \warusiki=0 または \syou=0\\
\\
& \begin{align*}
\warusiki &= 0\\
x &= \dainyu
\end{align*}\\
\\
& \syou = 0\\
& \begin{align*}
x &= \dfrac{2 \pm \sqrt{4-4 \cdot (-2)}}{2}\\
&= \dfrac{2 \pm \sqrt{12}}{2}\\
&= \dfrac{2 \pm 2\sqrt{3}}{2}\\
&= \dfrac{2(1 \pm \sqrt{3})}{2} = 1 \pm \sqrt{3}\\
\end{align*}\\
\\
& したがって x = 1,\ 1\pm\sqrt{3}
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan組立除法
\def\va{1}
\def\vbf{}\def\vb{-3}
\def\vcf{+}\def\vc{0}
\def\vdf{+}\def\vd{2}
\def\wv{1}
\def\ws{x-1}
\def\kb{1}
\def\tbf{}\def\tb{-2}
\def\kc{-2}
\def\tcf{}\def\tc{-2}
\def\kd{-2}
\def\td{0}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{matrix}
\va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\
\colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\
& & & & \colMM{red}{\Darr}\\
\va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\
& \kb & \kc & \kd & \\\hline
\va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\
\colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\
\va x^2 & \tb x & \tc\\
\end{matrix}
\\
\begin{align*}\\
\colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\
& \ \colFR{black}{\scriptsize\bf 余} \ \td
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan↓この問題へのリンクはこちら（右クリックで保存）
【解答】
\def\siki{2x^3-3x^2}\def\teisuf{}\def\teisu{-4}
\def\yakusu{\pm1,\ \pm2,\ \pm4}
\def\testsiki{%
& \colMM{orange}{P(1) = 2-3-4=-5}\\
& \colMM{orange}{P(-1) = -2-3-4=-9}\\
& \colMM{orange}{P(2) = 16-12-4=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた！}
}
\def\dainyu{2}
\def\dainyusiki{%
\begin{align*}
P(\dainyu) &= 2 \cdot 2^3-3 \cdot 2^2-4\\
\colMM{green}{\Darr } &= 16-12-4\\
\colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが０・・・割り切れる！}
\end{align*}
}
\def\warusiki{x-2}
\def\syou{2x^2+x+2}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{align*}
& \colFB{orange}{下調べ}\\
& \ \colMM{orange}{P(k)=0\ になる\ k\ を探す！}\\
& \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入！}\\
\testsiki\\
\\
& P(x) = \siki\teisuf\teisu\ とすると\\
& \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\
& \dainyusiki\\
& \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\
& よって，P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\
& \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り０}\\
& P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\
\\
& P(x) = 0\ から\\
& \warusiki=0 または \syou=0\\
\\
& \begin{align*}
\warusiki &= 0\\
x &= \dainyu
\end{align*}\\
\\
& \syou = 0\\
& \begin{align*}
x &= \dfrac{-1 \pm \sqrt{1-4 \cdot 4}}{4}\\
&= \dfrac{-1 \pm \sqrt{-15}}{4}\\
&= \dfrac{-1 \pm \sqrt{15}\,i}{4}\\
\end{align*}\\
\\
& したがって x = -1,\ \dfrac{-1 \pm \sqrt{15}\,i}{4}
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan組立除法
\def\va{2}
\def\vbf{}\def\vb{-3}
\def\vcf{+}\def\vc{0}
\def\vdf{}\def\vd{-4}
\def\wv{2}
\def\ws{x-2}
\def\kb{4}
\def\tbf{+}\def\tb{1}
\def\kc{2}
\def\tcf{+}\def\tc{2}
\def\kd{4}
\def\td{0}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{matrix}
\va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\
\colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\
& & & & \colMM{red}{\Darr}\\
\va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\
& \kb & \kc & \kd & \\\hline
\va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\
\colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\
\va x^2 & \tb x & \tc\\
\end{matrix}
\\
\begin{align*}\\
\colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\
& \ \colFR{black}{\scriptsize\bf 余} \ \td
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan因数分解の問題をリサイクル！
次の３次方程式を解け。
↓この問題へのリンクはこちら（右クリックで保存）
【解答】
\def\siki{x^3+2x^2-x}\def\teisuf{}\def\teisu{-2}
\def\yakusu{\pm1,\ \pm2}
\def\testsiki{%
& \colMM{orange}{P(1) = 1+2-1-2=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた！}
}
\def\dainyu{1}
\def\dainyusiki{%
\begin{align*}
P(\dainyu) &= 1^3+2 \cdot 1^2 -1-2\\
\colMM{green}{\Darr } &= 1+2-1-2\\
\colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが０・・・割り切れる！}
\end{align*}
}
\def\warusiki{x-1}
\def\syou{x^2+3x+2}
\def\insubunkai{(x+1)(x+2)}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{align*}
& \colFB{orange}{下調べ}\\
& \ \colMM{orange}{P(k)=0\ になる\ k\ を探す！}\\
& \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入！}\\
\testsiki\\
\\
& P(x) = \siki\teisuf\teisu\ とすると\\
& \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\
& \dainyusiki\\
& \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\
& よって，P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\
& \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り０}\\
& \begin{align*}
P(x) &= (\warusiki)(\syou)\colMM{magenta}{+0}\\
&= (\warusiki)\insubunkai\\
\end{align*}\\
\\
& P(x) = 0\ から\\
& \warusiki=0{\scriptsize または }x+1=0{\scriptsize または }x+2=0\\
\\
& したがって x = \pm1,\ -2
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan組立除法
\def\va{1}
\def\vbf{+}\def\vb{2}
\def\vcf{}\def\vc{-1}
\def\vdf{}\def\vd{-2}
\def\wv{1}
\def\ws{x-1}
\def\kb{1}
\def\tbf{+}\def\tb{3}
\def\kc{3}
\def\tcf{+}\def\tc{2}
\def\kd{2}
\def\td{0}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{matrix}
\va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\
\colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\
& & & & \colMM{red}{\Darr}\\
\va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\
& \kb & \kc & \kd & \\\hline
\va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\
\colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\
\va x^2 & \tb x & \tc\\
\end{matrix}
\\
\begin{align*}\\
\colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\
& \ \colFR{black}{\scriptsize\bf 余} \ \td
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan↓この問題へのリンクはこちら（右クリックで保存）
【解答】
\def\siki{x^3-7x}\def\teisuf{}\def\teisu{-6}
\def\yakusu{\pm1,\ \pm2,\ \pm3,\ \pm6}
\def\testsiki{%
& \colMM{orange}{P(1) = 1-7-6=-12}\\
& \colMM{orange}{P(-1) = -1+7-6=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた！}
}
\def\dainyu{-1}
\def\dainyusiki{%
\begin{align*}
P(\dainyu) &= (-1)^3-7 \cdot (-1) -6\\
\colMM{green}{\Darr } &= -1+7-6\\
\colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが０・・・割り切れる！}
\end{align*}
}
\def\warusiki{x+1}
\def\syou{x^2-x-6}
\def\insubunkai{(x+2)(x-3)}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{align*}
& \colFB{orange}{下調べ}\\
& \ \colMM{orange}{P(k)=0\ になる\ k\ を探す！}\\
& \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入！}\\
\testsiki\\
\\
& P(x) = \siki\teisuf\teisu\ とすると\\
& \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\
& \dainyusiki\\
& \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\
& よって，P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\
& \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り０}\\
& \begin{align*}
P(x) &= (\warusiki)(\syou)\colMM{magenta}{+0}\\
&= (\warusiki)\insubunkai\\
\end{align*}\\
\\
& P(x) = 0\ から\\
& \warusiki=0{\scriptsize または }x+2=0{\scriptsize または }x-3=0\\
\\
& したがって x = -1,\ -2,\ 3
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan組立除法
\def\va{1}
\def\vbf{+}\def\vb{0}
\def\vcf{}\def\vc{-7}
\def\vdf{}\def\vd{-6}
\def\wv{-1}
\def\ws{x+1}
\def\kb{-1}
\def\tbf{}\def\tb{-1}
\def\kc{1}
\def\tcf{}\def\tc{-6}
\def\kd{6}
\def\td{0}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{matrix}
\va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\
\colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\
& & & & \colMM{red}{\Darr}\\
\va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\
& \kb & \kc & \kd & \\\hline
\va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\
\colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\
\va x^2 & \tb x & \tc\\
\end{matrix}
\\
\begin{align*}\\
\colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\
& \ \colFR{black}{\scriptsize\bf 余} \ \td
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan応用問題に登場！
次の３次方程式を解け。
↓この問題へのリンクはこちら（右クリックで保存）
【解答】
\def\siki{x^3+x}\def\teisuf{+}\def\teisu{10}
\def\yakusu{\pm1,\ \pm2,\ \pm5,\ \pm10}
\def\testsiki{%
& \colMM{orange}{P(1) = 1+1+10=12}\\
& \colMM{orange}{P(-1) = -1-1+10=8}\\
& \colMM{orange}{P(2) = 8+2+10=20}\\
& \colMM{orange}{P(-2) = -8-2+10=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた！}
}
\def\dainyu{-2}
\def\dainyusiki{%
\begin{align*}
P(\dainyu) &= (-2)^3+(-2)+10\\
\colMM{green}{\Darr } &= -8-2+10\\
\colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが０・・・割り切れる！}
\end{align*}
}
\def\warusiki{x+2}
\def\syou{x^2-2x+5}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{align*}
& \colFB{orange}{下調べ}\\
& \ \colMM{orange}{P(k)=0\ になる\ k\ を探す！}\\
& \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入！}\\
\testsiki\\
\\
& P(x) = \siki\teisuf\teisu\ とすると\\
& \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\
& \dainyusiki\\
& \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\
& よって，P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\
& \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り０}\\
& P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\
\\
& P(x) = 0\ から\\
& \warusiki=0 または \syou=0\\
\\
& \begin{align*}
\warusiki &= 0\\
x &= \dainyu
\end{align*}\\
\\
& \syou = 0\\
& \begin{align*}
x &= \dfrac{2 \pm \sqrt{4-4 \cdot 5}}{2}\\
&= \dfrac{2 \pm \sqrt{-16}}{2}\\
&= \dfrac{2 \pm \sqrt{16}\,i}{2}\\
&= \dfrac{2 \pm 4\,i}{2}\\
&= \dfrac{2(1 \pm 2\,i)}{2} = 1 \pm 2\,i\\
\end{align*}\\
\\
& したがって x = -2,\ 1\pm2\,i
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan組立除法
\def\va{1}
\def\vbf{+}\def\vb{0}
\def\vcf{+}\def\vc{1}
\def\vdf{+}\def\vd{10}
\def\wv{-2}
\def\ws{x+2}
\def\kb{-2}
\def\tbf{}\def\tb{-2}
\def\kc{4}
\def\tcf{+}\def\tc{5}
\def\kd{-10}
\def\td{0}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{matrix}
\va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\
\colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\
& & & & \colMM{red}{\Darr}\\
\va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\
& \kb & \kc & \kd & \\\hline
\va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\
\colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\
\va x^2 & \tb x & \tc\\
\end{matrix}
\\
\begin{align*}\\
\colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\
& \ \colFR{black}{\scriptsize\bf 余} \ \td
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan↓この問題へのリンクはこちら（右クリックで保存）
【解答】
\def\siki{x^3+x^2-4x}\def\teisuf{+}\def\teisu{6}
\def\yakusu{\pm1,\ \pm2,\ \pm3,\ \pm6}
\def\testsiki{%
& \colMM{orange}{P(1) = 1+1-4+6=4}\\
& \colMM{orange}{P(-1) = -1+1+4+6=10}\\
& \colMM{orange}{P(2) = 8+4-8+6=10}\\
& \colMM{orange}{P(-2) = -8+4+8+6=10}\\
& \colMM{orange}{P(3) = 27+9-12+6=30}\\
& \colMM{orange}{P(-3) = -27+9+12+6=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた！}
}
\def\dainyu{-3}
\def\dainyusiki{%
\begin{align*}
P(\dainyu) &= (-3)^3+(-3)^2-4 \cdot (-3)+6\\
\colMM{green}{\Darr } &= -27+9+12+6\\
\colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが０・・・割り切れる！}
\end{align*}
}
\def\warusiki{x+3}
\def\syou{x^2-2x+2}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{align*}
& \colFB{orange}{下調べ}\\
& \ \colMM{orange}{P(k)=0\ になる\ k\ を探す！}\\
& \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入！}\\
\testsiki\\
\\
& P(x) = \siki\teisuf\teisu\ とすると\\
& \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\
& \dainyusiki\\
& \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\
& よって，P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\
& \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り０}\\
& P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\
\\
& P(x) = 0\ から\\
& \warusiki=0 または \syou=0\\
\\
& \begin{align*}
\warusiki &= 0\\
x &= \dainyu
\end{align*}\\
\\
& \syou = 0\\
& \begin{align*}
x &= \dfrac{2 \pm \sqrt{4-4 \cdot 2}}{2}\\
&= \dfrac{2 \pm \sqrt{-4}}{2}\\
&= \dfrac{2 \pm \sqrt{4}\,i}{2}\\
&= \dfrac{2 \pm 2\,i}{2}\\
&= \dfrac{2(1 \pm i)}{2} = 1 \pm i\\
\end{align*}\\
\\
& したがって x = -3,\ 1\pm i
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan組立除法
\def\va{1}
\def\vbf{+}\def\vb{1}
\def\vcf{}\def\vc{-4}
\def\vdf{+}\def\vd{6}
\def\wv{-3}
\def\ws{x+3}
\def\kb{-3}
\def\tbf{}\def\tb{-2}
\def\kc{6}
\def\tcf{+}\def\tc{2}
\def\kd{-6}
\def\td{0}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{matrix}
\va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\
\colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\
& & & & \colMM{red}{\Darr}\\
\va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\
& \kb & \kc & \kd & \\\hline
\va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\
\colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\
\va x^2 & \tb x & \tc\\
\end{matrix}
\\
\begin{align*}\\
\colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\
& \ \colFR{black}{\scriptsize\bf 余} \ \td
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan↓この問題へのリンクはこちら（右クリックで保存）
【解答】
\def\siki{x^3-3x^2+x}\def\teisuf{+}\def\teisu{5}
\def\yakusu{\pm1,\ \pm5}
\def\testsiki{%
& \colMM{orange}{P(1) = 1-3+1+5=4}\\
& \colMM{orange}{P(-1) = -1-3-1+5=\colBX{mistyrose}{\color{red}$0$}}\colMM{red}{\bf 見つけた！}
}
\def\dainyu{-1}
\def\dainyusiki{%
\begin{align*}
P(\dainyu) &= (-1)^3-3(-1)^2+(-1)+5\\
\colMM{green}{\Darr } &= -1-3-1+5\\
\colMM{green}{\Darr } &= \colBX{mistyrose}{$0$}\ \colMM{red}{余りが０・・・割り切れる！}
\end{align*}
}
\def\warusiki{x+1}
\def\syou{x^2-4x+5}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{align*}
& \colFB{orange}{下調べ}\\
& \ \colMM{orange}{P(k)=0\ になる\ k\ を探す！}\\
& \colMM{orange}{ \Rightarrow 定数項の約数\ \yakusu\ を代入！}\\
\testsiki\\
\\
& P(x) = \siki\teisuf\teisu\ とすると\\
& \colMM{orange}{ \ \Darr\ 下調べで見つけた\ x = \dainyu\ を代入}\\
& \dainyusiki\\
& \colMM{green}{x = \dainyu\ \Rightarrow \ \warusiki = 0\ \Darr}\\
& よって，P(x)\ は\ \colFR{green}{$\warusiki$}\ を\colBX{mistyrose}{$因数にもち$}\\
& \colMM{magenta}{P(x) \div (\warusiki)\ を計算すると商が\ \syou 余り０}\\
& P(x) = (\warusiki)(\syou)\colMM{magenta}{+0}\\
\\
& P(x) = 0\ から\\
& \warusiki=0 または \syou=0\\
\\
& \begin{align*}
\warusiki &= 0\\
x &= \dainyu
\end{align*}\\
\\
& \syou = 0\\
& \begin{align*}
x &= \dfrac{4 \pm \sqrt{16-4 \cdot 5}}{2}\\
&= \dfrac{4 \pm \sqrt{-4}}{2}\\
&= \dfrac{4 \pm \sqrt{4}\,i}{2}\\
&= \dfrac{4 \pm 2\,i}{2}\\
&= \dfrac{2(2 \pm i)}{2} = 2 \pm i\\
\end{align*}\\
\\
& したがって x = -1,\ 2\pm i
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan組立除法
\def\va{1}
\def\vbf{}\def\vb{-3}
\def\vcf{+}\def\vc{1}
\def\vdf{+}\def\vd{5}
\def\wv{-1}
\def\ws{x+1}
\def\kb{-1}
\def\tbf{}\def\tb{-4}
\def\kc{4}
\def\tcf{+}\def\tc{5}
\def\kd{-5}
\def\td{0}
\newcommand\colNS[2]{\color{#1}#2\color{black}}
\newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}}
\newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}}
\newcommand\colBX[2]{\colorbox{#1}{#2}}
\newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}}
\newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}}
\begin{matrix}
\va x^3 & \vbf\vb x^2 & \vcf\vc x & \vdf\vd & \colMM{red}{\ws=0}\\
\colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{green}{\Darr} & \colMM{red}{x = \wv}\\
& & & & \colMM{red}{\Darr}\\
\va & \vb & \vc & \vd & \colBX{mistyrose}{$\wv$}\\
& \kb & \kc & \kd & \\\hline
\va & \tb & \tc & \colBX{bisque}{$\td$} & \colMM{orange}{\Leftarrow 余り}\\
\colMM{red}{\Darr} & \colMM{red}{\Darr} & \colMM{red}{\Darr}\\
\va x^2 & \tb x & \tc\\
\end{matrix}
\\
\begin{align*}\\
\colFB{red}{Answer} & \ \colFR{black}{\scriptsize\bf 商} \ \va x^2 \tbf\tb x \tcf\tc \\
& \ \colFR{black}{\scriptsize\bf 余} \ \td
\end{align*}
%1 orange,bisque
%2 green,palegreen
%3 magenta, violet
%4 deepskyblue, lightcyan
